Any subgroup of the isometry group of a Riemannian manifold acts on such a manifold in a natural way, giving rise to the concept of isometric action. When an isometric action has an orbit of codimension one, such an action is said to be of cohomogeneity one.

An interesting problem arises when one tries to understand cohomogeneity one actions on Riemannian manifolds with a large isometry group, such as symmetric spaces. In this talk I will explain what is known about cohomogeneity one actions on symmetric spaces of noncompact type, focusing on the results by Berndt and Tamaru, as well as on an ongoing joint work with J. Carlos Díaz Ramos, where we develop new ideas to address the classification problem.